The world of Zettai Karen and the Unlimited is a world where Espers have existed for a long time. The world's history is similar to ours but with significant differences.

Espers seem to have been feared outcasts or designated the status of mystics until about WW2. During that war many nations tried to incorporate Espers as part of their military, post war efforts continued to integrate Espers with normals. Japan founded BABEL with that goal in mind 50 years ago.

There are some differences in the countries too

America is Comerica
Germany is Jermany
England is Ingland
France is Franse

Espers are rated between 1 and 7

1 - so weak you might as well not even have a power

4 - this seems to be the point where someone can be considered dangerous

7 - WMD

Power Limiters do exist but current technology can only dampen an esper by about 3 levels. From what I've seen espers seem to be required to wear limiters in public places like schools

Most Espers only have 1 power, though multiple powers isn't uncommon. Outside of main characters in the series I think I've only seen 1 person with more than 1 power.

Synthetic Powers are a blending of different aspects pure powers to create unique and often bizarre powers like turning your body into string (apparently a mix of teleportation and psychometry) or turning things into Giant Robots (Psychokineisis + Hypnotism)

Psychic powers are hereditary

Rare Metal is something that only comes from a middle eastern nation called Impalahen. It is capable of enhancing and storing psychic energy. Japan has been on good terms with Impalahen since ww2 and Japan Esper tech has advanced considerably with the help of Rare Metal alloys.

major nations have created their own Esper divisions, a lot of other nations however still treat Espers as non-human beings and seek ways to maximize their use as tools or weapons (as seen in Episode 1 of Unlimited) Animal experimentation is also pretty common.

Outside of the world governments there are 3 organizations of relevance to Espers

PANDRA: Hyobu's organization which exists to protect Espers from exploitation by world governments. They are considered a terrorist organization but they have made diplomatic connections with some nations who provide them with sanctuary and covers.

The Normal People: An Anti-Esper terrorist organization. These people believe that Espers are monsters and need to be exterminated, while they have often been portrayed in a mildly comical fashion they have shown that they are more than willing to kill a 10 year old child just because he has powers. They simply do not see Espers as humans.

Black Phantom: A Criminal Organization that uses extensive brainwashing on Espers to create Assassins and Agents. Black Phantom's brainwashing was considered unbreakable until the development of BABEL's triple boost system.

The Children

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Hyobu will often make reference to "The Queen"

The Queen is a prophesy that was originally predicted by two of the worlds most powerful precogs in WW2. Eventually Humans and Espers will enter into a global conflict against each other

The Espers will be saved by 3 women; The Queen of Catastrophe, the Light Speed Goddess and the Untouchable Empress

The exact details of this prediction are still unclear in a lot of aspects but one key event is the Death of the Queen, an event Hyobu is determined to change.

Those saviors at present are 12 year old Girls. The BABEL special Esper team known as The Children. The only three level 7 Espers in Japan. Their respective powers are
The Queen = Psychokineisis
The Goddess = Teleportation
The Empress = Psychometry

BABEL's own precognitive division has given even odds that the Children will Save humanity or Destroy it.

was a child soldier in the esper division of the Japanese military during ww2 alongside Tsubomi Fujiko. Both of them were exceptionally gifted and possess multiple psychic powers

both of them are over 80 years old and have separate methods to prevent themselves from aging. Fujiko is the founder of BABEL and Hyobu's natural nemesis on account of how much she used to tease him when they were kids.